Covert Communication in Wireless Relay Networks Jinsong Hu∗ , Shihao Yan† , Xiangyun Zhou†, Feng Shu∗ , and Jiangzhou Wang‡
arXiv:1704.04946v1 [cs.IT] 17 Apr 2017
∗
School of Electronic and Optical Engineering, Nanjing University of Science and Technology, Nanjing, Jiangsu, China † Research School of Engineering, The Australian National University, Canberra, ACT, Australia ‡ School of Engineering and Digital Arts, University of Kent, Canterbury, Kent, U.K. Emails:{jinsong hu, shufeng}@njust.edu.cn, {shihao.yan, xiangyun.zhou}@anu.edu.au,
[email protected]
Abstract—Covert communication aims to shield the very existence of wireless transmissions in order to guarantee a strong security in wireless networks. In this work, for the first time we examine the possibility and achievable performance of covert communication in one-way relay networks. Specifically, the relay opportunistically transmits its own information to the destination covertly on top of forwarding the source’s message, while the source tries to detect this covert transmission to discover the illegitimate usage of the recourse (e.g., power, spectrum) allocated only for the purpose of forwarding source’s information. The necessary condition that the relay can transmit covertly without being detected is identified and the source’s detection limit is derived in terms of the false alarm and miss detection rates. Our analysis indicates that boosting the forwarding ability of the relay (e.g., increasing its maximum transmit power) also increases its capacity to perform the covert communication in terms of achieving a higher effective covert rate subject to some specific requirement on the source’s detection performance.
I. I NTRODUCTION Security and privacy are critical in existing and future wireless networks since a large amount of confidential information (e.g., location, credit card information, physiological information for e-health) is transferred over the open wireless medium [1]. Against this background, conventional cryptography and information-theoretic secrecy technologies have been developed to offer progressively higher levels of security by protecting the content of the message against eavesdropping [2]–[4]. However, these technologies cannot mitigate the threat to a user’s security and privacy from discovering the presence of the user or communication. Meanwhile, this strong security (i.e., hiding the wireless transmission) is desired in many application scenarios of wireless communications, such as covert military operations and avoiding to be tracked in vehicular ad hoc networks. As such, the hiding of communication termed covert communication or low probability of detection communication, which aims to shield the very existence of wireless transmissions against a warden to achieve security, has recently drawn significant research interests and is emerging as a cutting-edge technique in the context of wireless communication security [5], [6]. The fundamental limit of covert communication has been studied under various channel conditions, such as additive white Gaussian noise (AWGN) channel [7], binary symmetric channel √ [8], and discrete memoryless channel [9]. It is proved that O( n) bits of information can be transmitted to a legitimate receiver reliably and covertly in n channel uses as n → ∞. This means that the associated covert rate is zero due
√ to limn→∞ O( n)/n → 0. Following these pioneering works on covert communication, a positive rate has been proved to be achievable when the warden has uncertainty on his receiver noise power [10], [11], the warden does not know when the covert communication happens [12], or an uniformed jammer comes in to help [13]. Most recently, [14] has examined the impact of noise uncertainty on covert communication by considering two practical uncertainty models in order to debunk the myth of this impact. In addition, the effect of the imperfect channel state information and finite blocklength (i.e., finite n) on covert communication has been investigated in [15] and [16], respectively. In this work, for the first time we consider covert communication in the context of one-way relay networks. This is motivated by the scenario where the relay (R) tries to transmit its own information to the destination (D) on top of forwarding the information from the source (S) to D, while S forbids R’s transmission of its own message since the resource (e.g., power, spectrum) allocated to R by S is dedicated to be solely used on aiding the transmission from S to D. As such, R’s transmission of its own message to D should be kept covert from S, where S acts as the warden trying to detect this covert communication. We first identify the necessary condition that the covert transmission from R to D can possibly occur without being detected by S and then derive the detection limit at S in terms of the false alarm and miss detection rates under this condition. In addition, we analyze the achievable effective covert rate subject to a requirement on the detection performance at S. Our examination demonstrates a tradeoff between R’s ability to aid the transmission from S to D and R’s capability to conduct the covert communication. II. S YSTEM M ODEL A. Considered Scenario and Adopted Assumptions As shown in Fig. 1, in this work we consider a one-way relay network, in which S transmits information to D with the aid of R, since a direct link from S to D is not available. As mentioned in the introduction, S allocates some resource to R in order to seek its help to relay the message to D. However, in some scenarios R may intend to use this resource to transmit its own message to D as well, which is forbidden by S and thus should be kept covert from S. As such, in the considered system model S is also the warden to detect whether R transmits its own information to D when it is aiding the transmission from S to D.
S as a payback. In this work, we assume both Rsd and δ are predetermined, which leads to a predetermined Prmax .
hrs Source
hsr Phase 1:
Relay
hrd
Phase 2:
III. T RANSMISSION S TRATEGIES AT R ELAY In this section, we detail the transmission strategies of R when it does and does not transmit its own information to D. We also determine the condition that R can transmit its own message to D without being detected by S with probability one, in which the probability to guarantee this condition is also derived.
Destination
Fig. 1. Covert communication in one-way relay networks.
We assume the wireless channels within our system model are subject to independent quasi-static Rayleigh fading with equal block length and the channel coefficients are independent and identically distributed (i.i.d.) circularly symmetric complex Gaussian random variables with zero-mean and unitvariance. We also assume that each node is equipped with a single antenna. The channel from S to R is denoted by hsr and the channel from R to D is denoted by hrd . We assume R knows both hsr and hrd perfectly, while S only knows hsr and D only knows hrd . Considering channel reciprocity, we assume the channel from R to S (denoted by hrs ) is the same as hsr and thus perfectly known by S. We further assume that R operates in the half-duplex mode and thus the transmission from S to D occurs in two phases: phase 1 (S transmits to R) and phase 2 (R transmits to D). B. Transmission from Source to Relay (Phase 1) In phase 1, the received signal at R is given by p yr [i] = Ps hsr xb [i] + nr [i],
(1)
where Ps is the transmit power of source, xb is the transmitted signal by S satisfying E[xb [i]x†b [i]] = 1, i = 1, 2, . . . , n is the index of each channel use (n is the total number of channel uses in each phase), and nr [i] is the AWGN at relay with σr2 as its variance, i.e., nr [i] ∼ CN (0, σr2 ). In this work, we consider that R operates in the amplifyand-forward mode and thus R will forward a linearly amplified version of the received signal to D in phase 2. As such, the forwarded (transmitted) signal by R is given by p xr [i] = Gyr [i] = G( Ps hsr xb [i] + nr [i]), (2)
where the received signal is scaled by a scalar G. In order to guarantee the power constraint at R, the value of G must be p chosen such that E[xr [i]x†r [i]] = 1, which leads to G = 1/ Ps |hsr |2 + σr2 . In this work, we consider a fixed-rate transmission from S to D, in which this rate is denoted by Rsd . We also consider a maximum power constraint at R, i.e., Pr ≤ Prmax . As such, although R knows both hsr and hrd perfectly, transmission ∗ outage from S to D still incurs when Csd < Rsd , where ∗ Csd is the channel capacity from S to D for Pr = Prmax . Then, the transmission outage probability is given by δ = ∗ P[Csd < Rsd ]. In practice, the pair of Rsd and δ determines the specific aid (i.e., the value of Prmax ) required by S from R, which relates to the amount of resource allocated to R by
A. Relay’s Transmission without Covert Message In the case when relay does not transmit its own message (i.e., covert message) to Bob, it only transmit xr to D. Accordingly, the received signal at D is given by p yd [i] = Pr0 hrd xr [i] + nd [i] (3) p p p 0 0 = Pr Ghrd Ps hsr xb [i] + Pr Ghrd nr [i] + nd [i],
where Pr0 is the transmit power of xr at R in this case and nd [i] is the AWGN at D with σd2 as the variance, i.e., nd [i] ∼ CN (0, σd2 ). Accordingly, the signal-to-noise ratio (SNR) at the destination for xb is given by γd =
γ1 γ2 Ps |hsr |2 Pr0 |hrd |2 G2 = , Pr0 |hrd |2 G2 σr2 + σd2 γ1 + γ2 + 1
(4)
where γ1 , (Ps |hsr |2 )/σr2 and γ2 , (Pr0 |hrd |2 )/σd2 . For the fixed-rate transmission, R does not have to adopt the maximum transmit power for each channel realization in order to guarantee a specific transmission outage probability. When ∗ the transmission outage occurs (i.e., Csd < Rsd occurs), R 0 ∗ will not transmit (i.e., Pr = 0). When Csd ≥ Rsd , R only has to ensure Csd = Rsd , where Csd = 1/2 log2 (1 + γd ). Then, ∗ following (4) the transmit power of R when Csd ≥ Rsd is 0 2 2 given by Pr = µσd /|hrd| , where µ,
(Ps |hsr |2 + σr2 )(22Rsd − 1) . [Ps |hsr |2 − σr2 (22Rsd − 1)]
(5)
Noting γd < γ1 , we have 1/2 log2 (1+γ1 ) > Rsd when Csd = Rsd . As such, µ given in (5) is nonnegative. Following (4), we ∗ note that Csd ≥ Rsd requires |hrd |2 ≥ µσd2 /Prmax . As such, the transmit power of R without covert message is given by µσd2 , |h |2 ≥ µσd2 , rd |hrd |2 Prmax 0 Pr = (6) µσd2 0, . |hrd |2 < max Pr
B. Relay’s Transmission with Covert Message
In the case when R transmits the covert message to D on top of forwarding xb , the received signal at D is given by p p p yd [i] = Pr1 Ghrd Ps hsr xb [i] + P∆ hrd xc [i] p (7) + Pr1 Ghrd nr [i] + nd [i].
where Pr1 is the transmit power of R to forward xb under this case and P∆ is the power of R used to transmit the covert message xc satisfying E[xc [i]x†c [i]] = 1. In this work, we assume that P∆ is fixed for all channel realizations. In
general, the transmit power of a covert message is significantly less than that of the forwarded message, i.e., P∆ Prmax − P∆ and sets Pr1 the same as Pr0 given in (6). This is due to the fact that S knows hrs and it can detect with probability one when the total transmit power of R is greater than Prmax . Then, the transmit power of xr under this case is given by
Pr1 = µP∆2 + µσd |hrd |2 , 0,
(10) µσd2 |hrd |2 ,
µσd2 |hrd | ≥ P max −(µ+1)P , ∆ r µσd2 µσd2 2 Prmax ≤ |hrd | < Prmax −(µ+1)P∆ , µσd2 . |hrd |2 < P max r 2
As per (10), we note that R also does not transmit covert message when it cannot support the transmission from S to D (i.e., when |hrd |2 < µσd2 /Prmax ). This is due to the fact that a transmission outage occurs when |hrd |2 < µσd2 /Prmax and D will request a retransmission from S, which enables S to detect R’s covert transmission with probability one if this happens. In summary, S cannot detect R’s covert transmission with probability one (R could possibly transmit covert message without being detected) only when the condition |hrd |2 ≥ µσd2 /[Prmax − (µ + 1)P∆ ] is guaranteed. We denote this necessary condition for covert communication as C. Considering Rayleigh fading, the cumulative distribution function (cdf) of |hrd |2 is given by F|hrd |2 (x) = 1 − e−x and thus the probability that C is guaranteed is given by µσd2 Pc = exp − max . (11) Pr − (µ + 1)P∆ We note that Pc is a monotonic decreasing function of P∆ , which indicates that the probability that R can transmit covert message (without being detected with probability one) decreases as P∆ increases. Following (9) and noting Pr1 + P∆ ≤ Prmax , we have Prmax > (µ + 1)P∆ and thus 0 ≤ Pc ≤ 1.
IV. B INARY D ETECTION AT S OURCE In this section, we first present the detection strategy adopted by S (i.e., Source) and then analyze the associated detection performance in terms of the false alarm and miss detection rates. Finally, the optimal detection threshold at S that minimizes the total error rate is examined. A. Binary Hypothesis Test In phase 2 when R transmits to D, S is to detect whether R transmits the covert message xc on top of forwarding S’s message xb to D. In this section, we only focus on the case when C is guaranteed since R never transmits covert message when C is not met, as discussed in Section III-B. R does not transmit xc in the null hypothesis H0 while it does in the alternative hypothesis H1 . Then, the received signal at S in phase 2 is given by ( p Pr0 hrs xr [i] + ns [i], p ys [i] = p Pr1 hrs xr [i] + P∆ hrs xc [i]+ns [i],
H0 ,
H1 ,
(12)
where ns [i] is the AWGN at S with σs2 as its variance. We note that neither Pr0 nor Pr1 is known at S since it does not know hrd , while the statistics of Pr0 and Pr1 are known since the distribution of hrd is publicly known. The ultimate goal of S is to detect whether ys comes from H0 or H1 in one fading block. As proved in [15], the optimal decision rule that minimizes the total error rate at S is given by D1
T ≷ τ,
(13)
D0
P where T = 1/n ni=1 |ys [i]|2 , τ is a predetermined threshold, D1 and D0 are the binary decisions that infer whether R transmits covert message or not, respectively. In this work, we consider infinite blocklength, i.e., n → ∞. As such, we have ( Pr0 |hrs |2 + σs2 , H0 , T= (14) 1 2 2 2 Pr |hrs | + P∆ |hrs | +σs , H1 . B. False Alarm and Miss Detection Rates In this subsection, we derive S’s false alarm rate, i.e., P(D1 |H0 ), and miss detection rate, i.e., P(D0 |H1 ). Theorem 1: When the condition C is guaranteed, the false alarm and miss detection rates at S are derived as τ < σs2 , 1, 1 − κ1 , σs2 ≤ τ ≤ ρ1 , (15) PF A = 0, τ > ρ1 , τ < ρ2 , 0, κ 2 , ρ2 ≤ τ ≤ ρ3 , PMD = (16) 1, τ > ρ3 ,
where ρ1 , [Prmax − (µ + 1)P∆ ]|hrs |2 + σs2 , ρ2 , (µ + 1)P∆ |hrs |2 + σs2 ,
ρ3 , Prmax |hrs |2 + σs2 , 1 |hrs |2 κ1 (τ ) , exp µσd2 , − Prmax − (µ + 1)P∆ τ − σs2 |hrs |2 1 . − κ2 (τ ) , exp µσd2 Prmax − (µ + 1)P∆ τ − ρ2 Proof: Considering the maximum power constraint at R under H0 (i.e., Pr0 ≤ Prmax ) and following (6), (13), and (14), the false alarm rate under the condition C is given by µσd2 2 2 PF A = P |hrs | + σs ≥ τ C (17) |hrd |2 1, τ < σs2 , 2 2 |h |2 µσd µσ rs P P max −(µ+1)P ≤|hrd |2 ≤ τd−σ2 ∆ = r s , σs2 ≤ τ ≤ ρ1 , Pc 0, τ > ρ1 .
Then, substituting F|hrd |2 (x) = 1 − e−x into the above equation (hrs is perfectly known by S and thus it is not a random variable here) we achieve the desired result in (15). We first clarify that we have ρ2 < ρ3 due to Prmax > (µ + 1)P∆ as discussed after (11). Then, considering the maximum power constraint at R under H1 (i.e., Pr1 + P∆ ≤ Prmax ) and following (10), (13), and (14), the miss detection rate under the condition C is given by µσd2 2 2 PMD = P + (1 + µ)P∆ |hrs | + σs < τ C |hrd |2 0, τ < ρ2 , 2 |h |2 µσd rs P |hrd |2 ≥ τ −(µ+1)P 2 2 = (18) ∆ |hrs | −σs , ρ2 ≤ τ ≤ ρ3 , Pc 1, τ > ρ3 .
Then, substituting F|hrd |2 (x) = 1 − e−x into the above equation we achieve the desired result in (16). We note that the false alarm and miss detection rates given in Theorem 1 are functions of the threshold τ and we examine how S sets the value of it in order to minimize its total error rate. Specifically, the total error rate of the detection at S is defined as ξ , PF A + PMD ,
(19)
which is used to measure the detection performance at S. C. Optimization of the Detection Threshold at Source In this subsection, we examine how S optimally sets the value of τ aiming to minimize ξ. To this end, we first determine a preliminary constraint on P∆ and the bounds on the optimal τ in the following theorem. Theorem 2: R’s transmit power of the covert message P∆ should satisfy u P∆ ≤ P∆ , Prmax /[2(µ + 1)]
(20)
in order to guarantee ξ > 0 and when (20) is guaranteed the optimal τ (τ ∗ ) at S that minimizes ξ should satisfy ρ2 ≤ τ ∗ ≤ ρ1 . Proof: When ρ1 < ρ2 that requires P∆ > Prmax /[2(µ + 1)] as per Theorem 1, following (15) and (16), we have 1, τ ≤ σs2 , 1 − κ1 (τ ), σs2 < τ < ρ1 , 0, ρ1 ≤ τ ≤ ρ2 , (21) ξ= κ (τ ), ρ < τ < ρ , 2 2 3 1, τ ≥ ρ3 .
This indicates that S can simply set τ ∈ [ρ1 , ρ2 ] to ensure ξ = 0 when P∆ > Prmax /[2(µ + 1)], i.e., S can detect the covert transmission with probability one. As such, P∆ should satisfy (20) in order to guarantee ξ > 0. When P∆ ≤ Prmax /[2(µ + 1)], i.e., ρ2 < ρ1 , following (15) and (16), we have 1, τ ≤ σs2 , σs2 < τ ≤ ρ2 , 1 − κ1 (τ ), 1 − κ1 (τ ) + κ2 (τ ), ρ2 < τ < ρ1 , (22) ξ= κ (τ ), ρ ≤ τ < ρ , 2 1 3 1, τ ≥ ρ3 ,
due to ρ3 > ρ1 . Obviously, the optimal value of τ cannot satisfy τ ≤ σs2 or τ ≥ ρ3 . For σs2 < τ ≤ ρ2 , we derive the first derivative of ξ with respect to τ as µσ 2 |hrs |2 ∂(ξ) = − d 2 2 κ1 < 0. ∂τ (τ − σs )
(23)
This demonstrates that ξ is a decreasing function of τ and thus we would have τ ∗ = ρ2 when σs2 < τ ≤ ρ2 . For ρ1 ≤ τ < ρ3 , we derive the first derivative of ξ with respect to τ as ∂(ξ) µσd2 |hrs |2 = 2 κ2 > 0. ∂τ [τ − (µ + 1)P∆ |hrs |2 − σs2 ]
(24)
This proves that ξ is an increasing function of τ and we would have τ ∗ = ρ1 when ρ1 ≤ τ < ρ3 . Noting that ξ is a continuous function of τ , we can conclude that τ ∗ should satisfy ρ2 ≤ τ ∗ ≤ ρ1 , no mater what is the value of ξ for ρ2 < τ < ρ1 . The lower and upper bounds on τ ∗ given in Theorem 2 significantly facilitate the numerical search for τ ∗ at S. Then, following Theorem 2 and (22), τ ∗ can be obtained through τ ∗ = argmin [1 − κ1 (τ ) + κ2 (τ )].
(25)
ρ2 ≤τ ≤ρ1
Substituting τ ∗ into (22), we obtain the minimum value of ξ as ξ ∗ = 1 − κ1 (τ ∗ ) + κ2 (τ ∗ ). V. O PTIMIZATION
OF
E FFECTIVE C OVERT R ATE
In this section, we examine the effective covert rate achieved in the considered system subject to a covert requirement.
1
A. Effective Covert Rate
As such, following (10) the SINR for xc is
P∆ (η|hsr |2 + 1)|hrd |2 , γc = µP∆ |hrd |2 + (η|hsr |2 + µ + 1)σd2
β1 , [η|hsr |2 + (µ + 1)](Prmax − P∆ )σd2 , η|hsr |2 + (µ + 1) 2 β2 , + µ P∆ σd2 , [Prmax − (µ + 1)P∆ ]−1 α1 , P∆ [η|hsr |2 + (µ + 1)][Prmax − (µ + 1)P∆ ],
0.4 0.3 PF A PM D ξ
0.2 0.1 0
2
4
6
8
10
12
14
τ
Fig. 2. PF A , PM D , and ξ versus different values of the threshold τ , where Ps = Prmax = 10 dB, σs2 = σr2 = σd2 = 0 dB, P∆ = 0.5, Rsd = 1, and |hsr |2 = |hrs |2 = 1.
B. Maximization of Rc with the Covert Constraint A covert transmission normally requires ξ ≥ 1 − ǫ, where ǫ ∈ [0, 1] is predetermined to specify the covert constraint. In practice, it is impossible to know ξ at R since the threshold τ adopted by S is unknown. In this work, we consider the worstcase scenario where τ is optimized at S to minimize ξ. As such, the covert constraint can be rewritten as ξ ∗ ≥ 1−ǫ. Then, following Theorem 2 the optimal value of P∆ that maximizes Rc subject to this constraint can be obtained through
s.t.
(29)
d max −(µ+1)P Pr ∆
a
0.5
(31)
u 0≤P∆ ≤P∆
Proof: A positive covert rate is only achievable under the condition C and thus Rc is given by Z ∞ Rc f (|hrd |2 )d|hrd |2 Rc = µσ2 1 µσd2 × exp − max ln 2 Pr − (µ + 1)P∆ Z ∞ β1 + α1 x e−x dx, ln β2 + α2 x 0
0.6
∗ P∆ = argmax Rc
α2 , µP∆ [Prmax − (µ + 1)P∆ ],
a
0.7
(27)
where
=
0.8
0
where η , Ps /σr2 . Then, the covert rate achieved by R is Rc = log2 (1+γc ). We next derive the effective covert rate, i.e., averaged Rc over all realizations of |hrd |2 , in the following theorem. Theorem 3: The achievable effective covert rate Rc by R is derived as a function of P∆ given by 1 µσd2 Rc = × exp − max ln 2 Pr − (µ + 1)P∆ β1 β2 β2 β1 β1 α1 α2 ln + e Ei − − e Ei − , (28) β2 α2 α1
and the exponential integral function Ei(·) is given by Z ∞ −t e Ei(x) = − dt, [x < 0]. −x t
0.9
Probability
As discussed in Section III-B, R can only transmit the covert message without being detected by S with probability one under the condition C. As such, a positive covert rate is only achievable under this condition. When the covert message is transmitted by R, D first decodes xb and subtracts the corresponding component from its received signal yd given in (7). Then, the effective received signal used to decode the covert message xc is given by p p ˜ d [i] = P∆ hrd xc [i] + Pr1 hrd Gnr [i] + nd [i]. (26) y
(30)
where = is achieved by exchanging variables (i.e., setting x = |hrd |2 −µσd2 /[Prmax − (µ + 1)P∆ ]). We then solve the integral in (30) with the aid of [17, Eq. (4.337.1)] and achieve the result given in (28). Based on Theorem 3, we note that Rc is not an increasing function of P∆ , since as P∆ increases Rc increases but Pc (i.e., the probability that the condition C is guaranteed) decreases.
ξ ∗ ≥ 1 − ǫ.
We note that this is a two-dimensional optimization problem that can be solved by efficient numerical searches. Specifically, for each given P∆ , ξ ∗ should be obtained based on (25) where τ ∗ is also numerically searched. We note that the numerical ∗ search of P∆ and τ ∗ is efficient since their lower and upper bounds are explicitly given. VI. N UMERICAL R ESULTS In this section, we first examine the detection performance at S (i.e., Source) under the condition C. Then, the impact of some system parameters on the achievable effective covert rate subject to a specific covert constraint is investigated. In Fig. 2, we plot the false alarm rate PF A , miss detection rate PMD , and total error rate ξ versus the threshold τ , in which the adopted system parameters guarantee condition C u and P∆ ≤ P∆ . As expected, we observe that ξ > 0 due u to the guaranteed condition C and P∆ ≤ P∆ , which means that covert transmission is possible (not being detected with probability one) under this condition. We observe that the minimum value of ξ is achieved when ρ2 ≤ τ ≤ ρ1 , which verifies the correctness of our Theorem 2. In Fig. 3, we plot the minimum total error rate ξ ∗ versus the covert transmit power P∆ , which is achieved through
1 Prmax Prmax Prmax Prmax
0.9
0.5
10 dB 8 dB 6 dB 4 dB
0.7 0.6
ξ∗
0.15
0.4
Effective covert rate
0.8
= = = =
0.5 0.4 0.3 0.2
0.1 Rsd = 4.0 0.3 0.05 0.2
0
0.1
0.2 Rsd = 4.3
0.1
|hsr |2 = 1.2
0.1 0
|hsr |2 = 1.0 0
0.2
0.4
0.6
0.8
1
P∆
0
0
0.2
0.4
0.6
0.8
1
P∆
Fig. 3. ξ ∗ versus P∆ with different value of Rsd , where Ps = 10 dB, σr2 = σd2 = 0 dB, Rsd = 1, and |hsr |2 = |hrs |2 = 1.
Fig. 4. Rc versus P∆ with different value of |hsr |2 , where Ps = Prmax = 30 dB, σr2 = σd2 = 0 dB, and ǫ = 0.1 (ǫ is only for the red circles).
searching the optimal threshold τ ∗ as per (25). In this figure, we first observe that ξ ∗ monotonically decreases as P∆ increases, which demonstrates that the covert transmission becomes easier to be detected when more power is used. As such, the covert constraint ξ ∗ ≥ 1 − ǫ determines a maximum u possible value of P∆ , which is significantly less than P∆ since u we have ξ = 0 when P∆ = P∆ but we normally require ξ > 0.5 in practice [16]. This can facilitate the search of the optimal value of P∆ as per (31) by significantly reducing the feasible region of P∆ . We also observe that ξ ∗ increases as Prmax increases. This shows that covert transmission becomes easier (i.e., the detection probability of covert transmission at S 1 − ξ ∗ decreases) as the desired performance of the normal transmission increases (i.e., the transmission outage probability decreases as Prmax increases for a fixed Rsd ).
This work examined covert communication in one-way relay networks over Rayleigh fading channels. Specifically, we analyzed S’s detection limit of the covert transmission from R to D in terms of the total error rate. We also determined the maximum achievable effective covert rate subject to ξ ∗ ≥ 1−ǫ. Our examination shows that covert communication in the considered relay networks is feasible and a tradeoff between the achievable effective covert rate and R’s performance of aiding the transmission from S to D exists.
VII. C ONCLUSION
In Fig. 4, we plot the effective covert rate Rc versus P∆ , in which we also show the maximum possible value of P∆ determined by the covert constraint ξ ∗ ≥ 1 − ǫ (denoted by ǫ P∆ and marked by red circle in this figure). We first observe that Rc may not be a monotonically increasing function of P∆ without the constraint ξ ∗ ≥ 1 − ǫ. This is due to the fact that as P∆ increases the probability to guarantee the condition C (i.e., Pc ) decreases while the covert rate Rc increases. In addition, we observe that Rc without ξ ∗ ≥ 1 − ǫ increases as |hsr |2 increases. This is due to the fact that as |hsr |2 increases µ as given in (5) decreases, which leads to that Pc increases, i.e., the probability that R can conduct covert transmission increases (although the covert rate Rc does not ǫ change). Finally, we observe that P∆ increases as well when 2 |hsr | increases. As such, following the last two observations we can conclude that the achievable effective covert rate with the constraint ξ ∗ ≥ 1 − ǫ increases as |hsr |2 increases. Intuitively, this is due to that as |hsr |2 increases R has a higher chance to support the transmission of xb and perform covert transmission, resulting in that from S’s point of view the possible transmit power range of R used to transmit xb increases (i.e., transmit power uncertainty increases).
ACKNOWLEDGMENTS This work was supported by the Australian Research Council’s Discovery Projects (DP150103905), the National Natural Science Foundation of China (No. 61472190), the Open Research Fund of National Key Laboratory of Electromagnetic Environment, China Research Institute of Radiowave Propagation (No. 201500013), and the open research fund of National Mobile Communications Research Laboratory, Southeast University, China (No. 2013D02). R EFERENCES [1] M. Bloch and J. Barros, Physical-Layer Security: From Information Theory to Security Engineering, 1st ed. Cambridge, UK: Cambridge University Press, 2011. [2] S. Yan, X. Zhou, N. Yang, B. He, and T. D. Abhayapala, “Artificialnoise-aided secure transmission in wiretap channels with transmitter-side correlation,” IEEE Trans. Wireless Commun., vol. 15, no. 12, pp. 8286– 8297, Dec. 2016. [3] J. Hu, F. Shu, and J. Li, “Robust synthesis method for secure directional modulation with imperfect direction angle,” IEEE Commun. Lett., vol. 20, no. 6, pp. 1084–1087, Jun. 2016. [4] J. Hu, S. Yan, F. Shu, J. Wang, J. Li, and Y. Zhang, “Artificial-noiseaided secure transmission with directional modulation based on random frequency diverse arrays,” IEEE Access, vol. 5, pp. 1658–1667, Jan. 2017. [5] B. A. Bash, D. Goeckel, D. Towsley, and S. Guha, “Hiding information in noise: fundamental limits of covert wireless communication,” IEEE Commun. Mag., vol. 53, no. 12, pp. 26–31, Dec. 2015. [6] M. Bloch, “Covert communication over noisy channels: A resolvability perspective,” IEEE Trans. Inf. Theory, vol. 62, no. 5, pp. 2334–2354, May 2016. [7] B. A. Bash, D. Goeckel, and D. Towsley, “Limits of reliable communication with low probability of detection on AWGN channels,” IEEE J. Sel. Areas Commun., vol. 31, no. 9, pp. 1921–1930, Sep. 2013.
[8] P. H. Che, M. Bakshi, and S. Jaggi, “Reliable deniable communication: Hiding messages in noise,” in Proc. IEEE Int. Symp. Inf. Theory, Jul. 2013, pp. 2945–2949. [9] L. Wang, G. W. Wornell, and L. Zheng, “Fundamental limits of communication with low probability of detection,” IEEE Trans. Inf. Theory, vol. 62, no. 6, pp. 3493–3503, Jun. 2016. [10] S. Lee, R. J. Baxley, M. A. Weitnauer, and B. T. Walkenhorst, “Achieving undetectable communication,” IEEE J. Sel. Topics Signal Process., vol. 9, no. 7, pp. 1195–1205, Oct. 2015. [11] D. Goeckel, B. A. Bash, S. Guha, and D. Towsley, “Covert communications when the warden does not know the background noise power,” IEEE Commun. Lett., vol. 20, no. 2, pp. 236–239, Feb. 2016. [12] B. A. Bash, D. Goeckel, and D. Towsley, “LPD communication when the warden does not know when,” in Proc. IEEE Int. Symp. Inf. Theory, Jul. 2014, pp. 606–610. [13] T. V. Sobers, B. A. Bash, D. Goeckel, S. Guha, and D. Towsley, “Covert communication with the help of an uninformed jammer achieves positive rate,” in Proc. Asilomar Conf. on Signals, Syst., and Comput., Nov. 2015, pp. 625–629. [14] B. He, S. Yan, X. Zhou, and V. K. N. Lau, “On covert communication with noise uncertainty,” IEEE Commun. Lett., vol. 21, no. 4, pp. 941–944, Apr. 2017. [15] K. Shahzad, X. Zhou, and S. Yan, “Covert communication in fading channels under channel uncertainty,” in Proc. IEEE VTC Spring, Jun. 2017, pp. 1–5, arXiv:1703.02169. [16] S. Yan, B. He, Y. Cong, and X. Zhou, “Covert communication with finite blocklength in AWGN channels,” in Proc. IEEE ICC, May 2017, pp. 1–6, arXiv:1701.08891. [17] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 7th ed. San Diego, CA, USA: Academic Press, 2007.
